Wednesday 29 May 2024

On This Day in Math - May 29


No matter how correct a mathematical theorem
may appear to be, one ought never to be satisfied
that there was not something imperfect
about it until it also gives
the impression of being beautiful.
~ George Boole

The 149th day of the year; There are 149 ways to put 8 queens on a 7-by-7 chessboard so that each queen attacks exactly one other queen. *Prime Curios

also 149 = 62 + 72 + 82.(note that the digits 1, 4, 9 are squares also)

And Derek Orr noted that the sum of the digits of 149, \(1 + 4 + 9 = 14 = 1^2 + 2^2 + 3^2 \)

149 is the smallest 3-digit prime with distinct digits in each position such that inserting a zero between any two digits creates a new prime (that is, 1049 & 1409 are both prime).

149 is the 35th prime number, and a twin prime with 151.

149 is an Emirp since 941, its reversal, is also a prime.

149 in binary is 10010101. The zeros are in prime positions 2, 3, 5, and 7, when read left-to-right. These are the four single digit prime numbers.*Prime Curios

149 is a strictly non-palindromic number, it is not a palindrome in any base from 2 to 147.

149 is a full reptend prime, its reciprocal is 148 digits long, 1/149 repeats 0067114093959731543624161073825503355704697986577181208053691275167785234899328859060402684563758389261744966442953020134228187919463087248322147651 indefinitely.


1733  Euler names (or mis-names) the "Pell Equations" and gives a method of multiple solutions. "Euler’s first excursion into Pell’s equation was his 1732 paper E-29, bearing a title that translates
as “On the solution of problems of Diophantus about integer numbers.” The main result of this paper is to show how certain quadratic Diophantine equations can be reduced to the Pell equation. In particular, he shows that if we can find a solution to the Diophantine equation \(y^2 = an^2 + bn + c \) and we can find solutions to the Pell equation, \(q^2 = ap^2 +1\), then we can use the solutions to the Pell equation to construct more solutions to the original Diophantine equation. He also shows how to use two solutions to a Pell equation to construct more solutions, and notes that solutions to a Pell equation give good rational approximations for the square root of a.  (Ed Sandifer, Euler and Pell, How Euler Did It. MAA) .
As early as 400 BC in India and Greece, mathematicians studied the numbers arising from the n = 2 case of Pell's equation, The first general method for solving the Pell's equation (for all N) was given by Bhāskara II in 1150, extending the methods of Brahmagupta. Called the chakravala (cyclic) method,  
Several European mathematicians rediscovered how to solve Pell's equation in the 17th century. Pierre de Fermat found how to solve the equation and in a 1657 letter issued it as a challenge to English mathematicians. In a letter to Kenelm Digby, Bernard Frénicle de Bessy said that Fermat found the smallest solution for N up to 150 and challenged John Wallis to solve the cases N = 151 or 313. Both Wallis and William Brouncker gave solutions to these problems, though Wallis suggests in a letter that the solution was due to Brouncker.

John Pell's connection with the equation is that he revised Thomas Branker's translation[14] of Johann Rahn's 1659 book Teutsche Algebra into English, with a discussion of Brouncker's solution of the equation. Leonhard Euler mistakenly thought that this solution was due to Pell, as a result of which he named the equation after Pell.



1832 Almost certain that he would die in a duel the next day, Evariste Galois first wrote “Letter to all Republicans,” and then wrote to a friend (Auguste Chevalier) describing his mathematics. It ended: “Eventually there will be, I hope, some people who will find it profitable to decipher this mess.” [Burton, History of Mathematics, p. 322]. See Smith, Source Book, pp. 278–285 for the letter. *VFR

The Galois memorial in the cemetery of Bourg-la-Reine. Évariste Galois was buried in a common grave and the exact location is unknown.
Galois Memorial

1898 the heirs of Alfred Nobel sign a "reconciliation agreement" so that lawyers and accountants can execute his will. The will's major bequest was to create the Nobel Prizes, but first, there were disputes to be settled.*TIS

1919 Proof of the general theory of relativity was observed during a total solar eclipse. São Tomé and Príncipe, officially the Democratic Republic of São Tomé and Príncipe, is a Portuguese-speaking island nation in the Gulf of Guinea, off the western equatorial coast of Central Africa. Príncipe was the site where astronomical observations of the total solar eclipse of 29 May 1919 confirmed Einstein's prediction of the curvature of light. The expedition was sponsored by the Royal Society and led by Sir Arthur Stanley Eddington. A solar eclipse permitted observation of the bending of starlight passing through the sun's gravitational field, as predicted by Einstein's theory of relativity. Separate expeditions of the Royal Astronomical Society travelled to Brazil and off the west coast of Africa. Both made measurements of the position of stars visible close to the sun during a solar eclipse. These observations showed that, indeed, the light of stars was bent as it passed through the gravitational field of the sun. The verification of predictions of Einstein's theory, proved during the solar eclipse was a dramatic landmark scientific event. *Wik

1957 Romania issued two stamps picturing a slide rule to publicize the 2nd Congress of the Society of Engineers and Technicians, which began in Bucharest on this day. [Scott #1159-60].
For the younger set... If you never used (saw) a slide rule, there is actually an online java app that you can simulate the use of one at this page.  The other instrument is a vernier caliper, used for measuring outside dimensions, inside diameter, and often a small depth measure.

2017 The Kepler conjecture, named after the 17th-century mathematician and astronomer Johannes Kepler, is a mathematical theorem about sphere packing in three-dimensional Euclidean space. It states that no arrangement of equally sized spheres filling space has a greater average density than that of the cubic close packing (face-centered cubic) and hexagonal close packing arrangements. The density of these arrangements is around 74.05%. (Time Hack, that is before 1700)
In 1998, Thomas Hales, following an approach suggested by Fejes Tóth (1953), announced that he had a proof of the Kepler conjecture. Hales' proof is a proof by exhaustion involving the checking of many individual cases using complex computer calculations. So now, 300 years have passed and we have a proof........Maybe.

In 2002  Referees said that they were "99% certain" of the correctness of Hales' proof, and the Kepler conjecture was accepted as a theorem for publication . 

In 2014, the Flyspeck project team, headed by Hales, announced the completion of a formal proof of the Kepler conjecture using a combination of the Isabelle and HOL Light proof assistants. In 2017, the formal proof was accepted by the journal Forum of Mathematics, Pi.  (time hack, 317+ years after Kepler, we have a proof.


1675 Humphry Ditton (May 29, 1675 – October 15, 1715) was born at Salisbury and died in London in 1715 at Christ's Hospital, where he was mathematical master. He does not seem to have paid much attention to mathematics until he came to London about 1705. W. W. Rouse Ball states that Ditton's 1706 book on fluxions occupied a place in English education equivalent to L'Hospital's book in France.

1859 John Walker (29 May 1781 – 1 May 1859) was an English inventor who invented the friction match.
He made them from small wooden sticks which he coated with sulphur, then tipped with a mixture of potassium chlorate, antimony sulphide and a binder of gum arabic. After searching for a suitable mixture with the intent of making a useful way to start a fire, he was successful on 27 Nov 1826. Beginning on 7 Apr 1827, he sold them in boxes of 50 for a shilling, with a folded slip of sandpaper as a striking surface. He called them Congreves, to honour Sir William Congreve, known for his invention of military rockets. He declined to patent the matches, yet was still able to make a comfortable income from them.  *TIS

He did not name the matches "Congreves" in honour of the inventor and rocket pioneer, Sir William Congreve as it is sometimes stated. The congreves were the invention of Charles Sauria, a French chemistry student at the time. He did not divulge the exact composition of his matches.

Two and a half years after Walker's invention was made public Isaac Holden arrived, independently, at the same idea of coating wooden splinters with sulphur. The exact date of his discovery, according to his own statement, was October 1829. Before that date Walker's sales-book contains an account of no fewer than 250 sales of friction matches, the first entry dated 7 April 1827.
 The credit for his invention was attributed only after his death.

1794 Johann Heinrich von Mädler  (29 May 1794, 14 Mar 1874 at age 79) German astronomer who (with Wilhelm Beer) published the most complete map of the Moon of the time, Mappa Selenographica, 4 vol. (1834-36). It was the first lunar map to be divided into quadrants, and it remained unsurpassed in its detail until J.F. Julius Schmidt's map of 1878. Mädler and Beer also published the first systematic chart of the surface features of the planet Mars (1830). *TIS

1882 Harry Bateman (29 May 1882 – 21 January 1946) He spent much of his life collecting special functions and integrals that solved partial differential equations. He kept the references on index cards stored in shoe boxes—eventually these began to crowd him out of his office. [DSB 1, 500] *SAU

With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poincare to a more expansive conformal group of spacetime leaving Maxwell's equations invariant. Moving to the US, he obtained a Ph.D. in geometry with Frank Morley and became a professor of mathematics at California Institute of Technology. There he taught fluid dynamics to students going into aerodynamics with Theodore von Karman. Bateman made a broad survey of applied differential equations in his Gibbs Lecture in 1943 titled, "The control of an elastic fluid".*Wik

1885 Finlay Freundlich (May 29, 1885 – July 24, 1964) was a distinguished German astronomer who worked with Einstein on measurements of the orbit of Mercury to confirm the general theory of relativity. He left Germany to avoid Nazi rule and became the Napier Professor of Astronomy at St Andrews.

1906 Gerrit Bol (May 29, 1906 in Amsterdam, Nov 1, 1989) was a Dutch mathematician, who specialized in geometry. He is known for introducing Bol loops in 1937, and Bol’s conjecture on sextactic points.
Bol earned his PhD in 1928 at Leiden University under Willem van der Woude. In the 1930s, he worked at the University of Hamburg on the geometry of webs under Wilhelm Blaschke and later projective differential geometry. In 1931 he earned a habilitation.
In 1942–1945 during World War II, Bol fought on the Dutch side, and was taken prisoner. On the authority of Blaschke, he was released. After the war, Bol became professor at the Albert-Ludwigs-University of Freiburg, until retirement there in 1971. *Wik

1911 George Szekeres (29 May 1911 – 28 August 2005) was a Hungarian-born mathematician who worked for most of his life in Australia on geometry and combinatorics. *SAU

Szekeres worked closely with many prominent mathematicians throughout his life, including Paul Erdős, Esther Szekeres (née Esther Klein), Paul Turán, Béla Bollobás, Ronald Graham, Alf van der Poorten, Miklós Laczkovich, and John Coates.

The so-called Happy Ending problem is an example of how mathematics pervaded George's life. During 1933, George and several other students met frequently in Budapest to discuss mathematics. At one of these meetings, Esther Klein proposed the following problem:

Given five points in the plane in general position, prove that four of them form a convex quadrilateral.

After allowing George, Paul Erdős, and the other students to scratch their heads for some time, Esther explained her proof. Subsequently, George and Paul wrote a paper (1935) that generalizes this result; it is regarded as one of the foundational works in the field of combinatorial geometry. Erdős dubbed the original problem the "Happy Ending" problem because it resulted in George and Esther's marriage in 1937.

George and Esther died within an hour of each other, on the same day, 28 August 2005, in Adelaide, Australia.*Wik

My story of the "Happy Ending" story is here.

1997 Chien-Shiung Wu (simplified Chinese: 吴健雄; traditional Chinese: 吳健雄; pinyin: Wú Jiànxióng, May 31, 1912 – February 16, 1997) was a Chinese American experimental physicist who made significant contributions in the field of nuclear physics. Wu worked on the Manhattan Project, where she helped develop the process for separating uranium metal into uranium-235 and uranium-238 isotopes by gaseous diffusion. She is best known for conducting the Wu experiment, which contradicted the hypothetical law of conservation of parity. This discovery resulted in her colleagues Tsung-Dao Lee and Chen-Ning Yang winning the 1957 Nobel Prize in physics, and also earned Wu the inaugural Wolf Prize in Physics in 1978. Her expertise in experimental physics evoked comparisons to Marie Curie. Her nicknames include "the First Lady of Physics", "the Chinese Madame Curie", and the "Queen of Nuclear Research".*Wik

1918 David Rees FRS (29 May 1918 – 16 August 2013) was a British professor of pure mathematics at the University of Exeter, having been head of the Mathematics / Mathematical Sciences Department at Exeter from 1958 to 1983. During the Second World War, Rees was active on Enigma research in Hut 6 at Bletchley Park.

Rees won a scholarship to Sidney Sussex College, Cambridge, supervised by Gordon Welchman and graduating in summer 1939. On completion of his education, he initially worked on semigroup theory; the Rees factor semigroup is named after him. He also characterised completely simple and completely 0-simple semigroups, in what is nowadays known as Rees's theorem. The matrix-based semigroups used in this characterisation are called Rees matrix semigroups.

Later in 1939, Welchman drafted Rees into Hut 6, Bletchley Park, for the war effort. He was credited with the first decode using the Herivel tip. He was subsequently seconded to the Enigma Research Section, where the Abwehr Enigma was broken, and later to the Newmanry, where the Colossus computer was built.

After the war, Rees was appointed an assistant lecturer at Manchester University in 1945 and a full lecturer at University of Cambridge in 1948. In 1949, he was a Fellow of Downing College.

At the behest of Douglas Northcott he switched his research focus to commutative algebra.[10] In 1954, in a joint paper with Northcott, Rees introduced the Northcott–Rees theory of reductions and integral closures, which has subsequently been influential in commutative algebra. In 1956 he introduced the Rees decomposition of a commutative algebra.

In 1958, Rees and his family moved to Exeter, where he had been appointed to the Chair of Pure Mathematics. In 1959, he was awarded a DSc by the University of Cambridge.

According to Craig Steven Wright, Rees was the third part of the Satoshi team that created Bitcoin.*Wik

1929 Günter Lumer (1929–2005) was a mathematician known for his work in functional analysis. He is the namesake of the Lumer–Phillips theorem on semigroups of operators on Banach spaces, and was the first to study L-semi-inner products. Born in Germany and raised in France and Uruguay, he spent his professional career in the United States and Belgium.
Following short-term positions at the University of California, Los Angeles and Stanford University, he joined the faculty at the University of Washington in 1961. He moved to the University of Mons-Hainaut in 1973, and then to the International Solvay Institutes for Physics and Chemistry in Brussels in 1999, where he remained until his death in 2005

1929 Peter Ware Higgs (29 May 1929 -  8 April 2024) is an English theoretical physicist, the namesake of the Higgs boson. In the late 1960s, Higgs and others proposed a mechanism that would endow particles with mass, even though they appeared originally in a theory - and possibly in the Universe! - with no mass at all. The basic idea is that all particles acquire their mass through interactions with an all-pervading field, called the Higgs field. which is carried by the Higgs bosons. This mechanism is an important part of the Standard Model of particles and forces, for it explains the masses of the carriers of the weak force, responsible for beta-decay and for nuclear reactions that fuel the Sun. The particle was discovered on 4 July 2012 at the Large Hadron Accelerator.

1957 Jean-Christophe Yoccoz ( May 29, 1957 -   )  French mathematician who was awarded the Fields Medal in 1994 for his work in dynamical systems. Such studies began with Poincaré about the turn of the 20th century, who considered the stability of the solar system. It evolves according to Newton's laws but will it remain stable or, might a planet be ejected from the system? The techniques apply also in biology, chemistry, mechanics, and ecology where stability is an issue. This work also produces aesthetically appealing objects, such as the Julia and Mandelbrot fractal sets. Yoccoz was primarily concerned with establishing criteria that gave precise bounds on the validity of stability theorems. A combinatorial method for studying the Julia and Mandelbrot sets was named "Yoccoz puzzles." *TIS


1660 Frans van Schooten (1615 in Leiden – 29 May 1660 in Leiden) was a Dutch mathematician who was one of the main people to promote the spread of Cartesian geometry. Van Schooten's father was a professor of mathematics at Leiden, having Christiaan Huygens, Johann van Waveren Hudde, and René de Sluze as students.
Van Schooten read Descartes' Géométrie (an appendix to his Discours de la méthode) while it was still unpublished. Finding it hard to understand, he went to France to study the works of other important mathematicians of his time, such as François Viète and Pierre de Fermat. When Frans van Schooten returned to his home in Leiden in 1646, he inherited his father's position and one of his most important pupils, Huygens.
Van Schooten's 1649 Latin translation of and commentary on Descartes' Géométrie was valuable in that it made the work comprehensible to the broader mathematical community, and thus was responsible for the spread of analytic geometry to the world. Over the next decade he enlisted the aid of other mathematicians of the time, de Beaune, Hudde, Heuraet, de Witt and expanded the commentaries to two volumes, published in 1659 and 1661. This edition and its extensive commentaries was far more influential than the 1649 edition. It was this edition that Gottfried Leibniz and Isaac Newton knew.
Van Schooten was one of the first to suggest, in exercises published in 1657, that these ideas be extended to three-dimensional space. Van Schooten's efforts also made Leiden the centre of the mathematical community for a short period in the middle of the seventeenth century. *Wik    Thony Christie (aka The Renaissance Mathematicus) sent me a comment to tell me that it was van Schooten who first used rectangular coordinates in his translations and extensions of Descartes Geometry.  The MAA Digital Library has seven images from van Schooten's "Exercitationes mathematicae". The copy was once the property of his student, Johann Hudde, and include problems from the book of another of his famous students, Christian Huygen's Ludo aleae.

Thony Christie added a note about van Schooten's contributions:
If you read La Géométrie you will search for rectangular co-ordinates in vain, Descartes did not use them. (Neither did Fermat who developed/invented algebraic geometry independently from Descarte). The first person to use them was van Schooten in his extended translation of Descartes work. (Thanks Thony)

1829 Sir Humphrey Davy (Baronet) (17 December 1778 – 29 May 1829) English chemist who discovered several chemical elements and compounds, invented the miner's safety lamp, and epitomized the scientific method. With appointment to the Pneumatic Institution to study the physiological effects of new gases, Davy inhaled gases (1800), such as nitrous oxide (laughing gas) and a nearly fatal inhalation of water gas, (a mixture of hydrogen and carbon monoxide). Davy discovered alkali metals, potassium and sodium, an isolation made with electric current for the first time (1807); as well as alkaline earth metals: calcium, strontium, barium, and magnesium (1808). He discovered boron at the same time as Gay-Lussac. He recognized chlorine as an element, which prior workers confused as a compound. *TIS Davy died in Switzerland in 1829 of heart disease inherited from his father's side of the family. He spent the last months of his life writing "Consolations In Travel", an immensely popular, somewhat freeform compendium of poetry, thoughts on science and philosophy (and even speculation concerning alien life) which became a staple of both scientific and family libraries for several decades afterward. He is buried in the Plainpalais Cemetery in Geneva.

1908 William Arnold Anthony (November 17, 1835 – May 29, 1908) was an American physicist  and electrical engineer who initiated and developed one of the first courses in electrical engineering in the U.S. (1883), while teaching in the Physics Department at Cornell University, Ithaca, N.Y. During 1872-75, Anthony, with the aid of student George Moler, built the first American Gramme dynamo for direct current, used to power arc lamps that lighted the Cornell campus, the first American electrical outdoor-lighting system. Anthony also built a mammoth tangent galvanometer, a device which utilized the earth's magnetic field for the measurement of current. He designed the dynamo for first underground electricity distributing system. Anthony contributed to development of gas-filled electric lamps.*TIS

1999 John Peter Louis Knopfmacher ( 20 January 1937 in Johannesburg – 29 May 1999 in Graz ) was a South African mathematician.*Wik

 John Knopfmacher studied accounting and then mathematics at the University of the Witwatersrand . While still a student, his first publication on perfect numbers appeared in the Mathematical Gazette in 1960. His son, Arthur, related:

"He related to me that as only a first year student, he became inspired by the famous problem of odd perfect numbers and derived what he believed to be a proof that none existed. One of his lecturers realised that while not a proof of this, it was in fact a new proof of the formula that describes all even perfect numbers, and this result became his first publication in a mathematics journal." *SAU

He received his bachelor's degree in 1958 and his master's degree in 1961. He then went to the University of Manchester , where he received his doctorate in 1965 under John Frank Adams ( Extensions in Varieties of Groups and Algebras ).  After returning (1965) to the University of Witwatersrand, he became a lecturer , in 1966 a senior lecturer, in 1971 a reader and associate professor, and in 1979 a professor. From 1984 to 1994 he was head of the mathematics department at his university and became City of Johannesburg professor. In 1992 he founded the Centre for Applicable Analysis and Number Theory at the university, which was named after him in 1999. He retired in 1997 and moved to Melbourne . He was most recently a visiting professor at the University of Graz .

He worked on algebra ( non-associative algebras, finite groups , Lie algebras ) and topology before turning to analytic number theory in 1970. In particular, he established an abstract (algebraic) approach to analytic number theory, which was also the title of his 1975 monograph. For this purpose, he developed the theory of arithmetic semigroups (arithmetic of free commutative semigroups with unity and real-valued multiplicative norm with some additional properties). The theory also allows, for example, consideration over finite fields.

He was married to Rose Hendler from 1959 until their divorce in 1991 and had three children. His son Arnold Knopfmacher is also a mathematician, with whom John Knopfmacher published around 35 joint works. 

In 1995 he received the Lifetime Achievement Medal of the South African Mathematical Society and was for many years the editor of its journal Quaestiones Mathematicae. In 1991 he was made a Fellow of the Royal Society of South Africa.*Wik

2005 Kazimierz Urbanik (February 5, 1930 – May 29, 2005) was a prominent member of the Polish School of Mathematics. He founded the journal Probability and Mathematical Statistics and served as rector of the University of Wrocław.

Urbanik began teaching at the University of Wrocław in 1956. By 1960, he was promoted to professor, and in 1965 he became a member of the Polish Academy of Sciences, becoming its youngest member. He was an invited speaker at the International Congress of Mathematicians in 1966. He directed the university's Institute of Mathematics for most of the years from 1967 to 1996, and was rector of the university from 1975 to 1981. In 1980, he founded the journal Probability and Mathematical Statistics, and became its first editor-in-chief.

His research contributions include over 180 papers. His work in probability theory included work on random variables in compact groups, connections between measurability and connectivity, generalized convolutions, and decomposability semigroups. He also studied stochastic processes, information theory, universal algebra, and functional analysis. He was the doctoral advisor of 17 students.

Credits :
*CHM=Computer History Museum
*FFF=Kane, Famous First Facts
*NSEC= NASA Solar Eclipse Calendar
*RMAT= The Renaissance Mathematicus, Thony Christie
*SAU=St Andrews Univ. Math History
*TIA = Today in Astronomy
*TIS= Today in Science History
*VFR = V Frederick Rickey, USMA
*Wik = Wikipedia
*WM = Women of Mathematics, Grinstein & Campbell

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